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Set Theory and the Foundation of Mathematics

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Set Theory and the Foundation of Mathematics Axiomatic set theory and the foundation of mathematics April { August 2019, Kobe 渕野 昌 (Saka´eFuchino) [email protected] main updates: 19.07.17(We09:24(JST)) 19.06.29(Sa00:35(JST)) Last updated on: 2021 年 04 月 08 日 (09:20 (JST))(0) Contents 1. Na¨ıve axiomatic set theory ................................................... 1 1.1. Zermelo Set Theory and the Axiom of Choice . 2 1.2. Zermelo-Fraenkel Set Theory . 4 2. Predicate logic ................................................................. 4 2.1. Predicate logic in meta-mathematics . 4 2.2. Axiomatic set theory over predicate logic . 9 2.3. The deduction system K∗ ................................................. 12 2.4. Predicate logic in Z and the Completeness Theorem . 18 2.5. Some semantic proofs of proof theoretic assertions . 18 3. Incompleteness Theorems and Speed-up Theorems ....................... 19 3.1. Ehrenfeucht-Mycielski Speed-up Theorem . 19 3.2. Diagonal lemma . 23 3.3. G¨odel'sSpeed-up Theorem . 29 References ........................................................................ 31 1 Na¨ıve axiomatic set theory naive In the following, The axiom system ZFC (Zermelo-Fraenkel Set Theory with Axiom of Choice) of set theory is formulated with certain redundancy. This formulation is chosen (0) The present text is a lecture note of the course I gave in the summer semester 2019 at Kobe university. The text is further extended and worked-out when I gave a similar course in Katowice in October 2019. The most up to date version of this text is downloadable as: http://fuchino.ddo.jp/kobe/logic-ss2019.pdf At present this text is still a work in progress. It will be constantly updated during and after the courses in 2019. Any comments and/or questions in connection with this text are appreciated, and are to be sent to the email address above. 1 on purpose so that some important subtheories of ZFC are subsets of the system. The axioms of ZFC are built on the single predicate \2" where \x 2 y" means x belongs to y as an element. We do not introduce a predicate for expressing \x is a set" since everything is a set in set theory.(1) If we say \for all x ..." in an axiom of set theory, what is meant is \for all set x, ...". 1.1 Zermelo Set Theory and the Axiom of Choice zermelo The following axioms make the axiom system of Zermelo Set Theory (Z) which is a subset of the axiom system of ZFC introduced in the next subsection. (Extensionality) If we have u 2 x if and only if u 2 y for any u, then x = y. (Empty Set) There is x such that u 62 x for any u. By the Axiom of Extensionality, the set x as in the Axiom of Empty Set exists uniquely. We denote this set by ;. (Pairing Axiom) For x and y there is z such that, for any u, u 2 z if and only if u = x or u = y. Again by the Axiom of Extensionality, the set z as in the Pairing Axiom for each x and y is unique. We shall denote this unieque z up to x and y by fx; yg. If x = y, we write fx; xg = fxg and call it the singleton x. (Axiom of Union) For any x, there is y such that, for any u, u 2 y if and only if u 2 z S for some z 2 x (notation: y = x). The next axiom is actually an axiom scheme in the sense that, for each property Φ, an instance of the axiom is made. (Axiom of Separation) If Φ(·) is a property expressed by using only \2" and \=" then for any x there is y such that u 2 y if and only if u 2 x and Φ(u) holds. The set y in the Axiom of Separation is often denoted by y = fu 2 x : Φ(u)g. S By Pairing axiom and Axiom of Union, we can construct fx; yg for any x and y. This set is denoted by x [ y. Clearly we have u 2 x [ y if and only if u 2 x or u 2 y for any u. (Axiom of Infinity) There is x such that 2 (2) (1.1) ; 2 x and y [ fyg 2 x for any any y 2 x. naive-a (Power-Set Axiom) For any x, there is y such that, for any u, u 2 y if and only if all elements of u are elements of x. The set y in the Power-set Axiom is denoted by y = P(x) and y is said to be a it power- set of x. The relation \u 2 y then u 2 x for all u" is denoted by y ⊆ x and y is said to be a subset of x. With this notation, we can write P(x) = fy : y ⊆ xg. The construction of power-set seen in this way is outside the scope of the Axiom of Separation since a set which would delimit the range of y in fy : y ⊆ xg (for example P(x)) can only be obtained using the Power-Set Axiom. Ordered pair of sets a and b can be treated as the following set. (1.2) ha; bi = ffag; fa; bgg. naive-0 . This definition is a reasonable one beacuse of: L-naive-0 Lemma 1.1 For any a, b and a0, b0 ha; bi = ha0; b0i if and only if a = a0 and b = b0. L-naive-1 Lemma 1.2 For any a and b the is c such that, for any u, u 2 c if and only if there are v 2 a and w 2 b such that u = hv; wi. Proof. Recalling (1.2), such c can be introduced as (1.3) fu 2 P(P(a [ b)) : u = hv; wi for some v 2 a and w 2 bg. naive-1 There is such a set because of the Axiom of Power-set and the Axiom of Separation. (Lemma 1.2) For a, b, the set c as in Lemma 1.2 is uniquely determined. We call such c a (Cartesian) product of a and b, and denote it by a × b. For a, and b, f ⊆ a × b is said to be a function from a to b, if, for any v 2 a, there is a unique w 2 b such that hv; wi 2 f (notation: f : a ! b). For v 2 a the unique w 2 b with hv; wi 2 f is denoted by f(v) = w and called the value of f at v. S (Axiom of Choice) For any x such that ; 62 x, there is a function f : x ! x such that f(u) 2 u for all u 2 x. (1) That is, except proper classes which are just reformulations of properties of sets and hence not objects in set theory. (2) Not that the set y [ fyg is the set of all elements of y and y itself. In [Zermelo 1908], the Axiom of Infinity was formulated with (1.1) replaced by \; 2 x and fyg 2 x for any y 2 x". The original form of the Axiom of Infinity is equivalent to our Axiom of Infinity in ZF but the equivalence is not provable in the Zermelo Set Theory Z. The formulation of the Axiom of Infinite with (1.1) is choosen here to make the collecgtion of all natural numbers ! a set under the standard definition of natural numers due to von Neumann possible in Z. 3 Axiom of Choice is abbreviated as AC. The axiom system obtained by adding the Axiom of Choice to the Zermelo Set Theory Z is denoted by ZC and called the Zermelo Set Theory with AC. 1.2 Zermelo-Fraenlel Set Theory zermelo-fraenkel The axiom system of Zermelo-Fraenlel Set Theory (ZF) is obtained by adding the follow- ing two axioms to the axiom systemo of Zermelo Set Theory. The first one of the two additional axioms is actually an axiom scheme like Axiom of Separation: (Replacement) For any x, if Φ(·; ·) is a property expressed by using only \2" and \=" such that, for any u 2 x there is the unique v such that Φ(u; v) holds, then there is y such that v 2 y if and only if Φ(u; v) for some u 2 x (notation: y = fv : Φ(u; v) for some u 2 xg). The significance of the next axiom of ZF becomes clear in connection with the theory of the transfinite induction. (Foundation) For any non empty x, there is y 2 x such that there is no y0 2 x with y0 2 y. 2 Predicate Logic pred-logic 2.1 Predicate logic in meta-mathematics meta The na¨ıve approach to the axiomatization of set theory we saw in the previous section was inacurate in many respects. One of the main problem was that we could not specify what are \set theoretic properties" in the definition of the Axiom of Separation and Axiom of Replacement. We want to re-introduce the axioms of set theory on the basis of the formal logic which will be introduced below. When we introduce the logic, we do it before introducing the axionms of set theory. We are in a world outside set theory. This means in particular that, at this stage, we cannot use the fictive notion of infinity, like the set of all natural numbers. If we write nevertheless some thing like \For n 2 N, ...", then this is merely abbreviation of the statement \For a concretely given expression (numeral) n of a natural number, ...".
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